Method for correcting pump model

ABSTRACT

A method for correcting pump model includes: obtaining a pump model for a pump, the pump model including a Q-P curve and a Q-H curve at each of a plurality of frequencies; at each frequency, determining a power error based on a zero-flow-rate power or an actual operating point of the pump under one of the frequencies; and determining a corrected pump model based on the pump model and the power error. The determination of the corrected pump model includes: at each frequency, determining a corrected Q-P curve to be the Q-P curve shifted by the power error; determining a Q-H-P surface based on the corrected Q-P curves and the Q-H curves at all the frequencies; at each frequency, determining a head error based on the surface and the power error; and at each frequency, determining a corrected Q-H curve to be the Q-H curve shifted by the head error.

RELATED APPLICATIONS

This application claims priority to Taiwan Application Serial Number 109145793, filed Dec. 23, 2020, the disclosure of which is incorporated herein by reference in its entirety.

BACKGROUND Technical Field

The present disclosure relates to a method for correcting pump model.

Description of Related Art

A pump is a device for moving fluid. Characteristics of a pump may be described by the pump characteristic curves (e.g., Q-P curves and Q-H curves, in which Q stands for flow rate, P stands for power and H stands for head). The pump characteristic curves may also be utilized to estimate the state of the pump. However, operating the pump could cause the characteristics of the pump to change (e.g., due to the wearing of the pump). In this case, directly using the pump characteristic curves to estimate the state of the pump would get inaccurate results. Therefore, it is necessary to correct the pump characteristic curves to reflect the change of pump characteristics, such that the estimated state of the pump can be more close to the actual state of the pump.

SUMMARY

In view of the foregoing, one of the objects of the present disclosure is to provide a method for correcting pump model to resolve the aforementioned problem.

A technical aspect of the present disclosure relates to a method for correcting pump model. The method includes: obtaining a pump model for a pump, the pump model including a Q-P curve (flow rate-power curve) and a Q-H curve (flow rate-head curve) at each of a plurality of frequencies for the pump; at each of the plurality of frequencies, determining a power error based on a zero-flow-rate power or an actual operating point of the pump at a first frequency of the plurality of frequencies; and determining a corrected pump model based on the pump model and the power error. The step of determining the corrected pump model based on the pump model and the power error includes: at each of the plurality of frequencies, determining a corrected Q-P curve to be the Q-P curve shifted by the power error; determining a Q-H-P surface based on the corrected Q-P curves and the Q-H curves at the plurality of frequencies; at each of the plurality of frequencies, determining a head error based on the Q-H-P surface and the power error; and at each of the plurality of frequencies, determining a corrected Q-H curve to be the Q-H curve shifted by the head error.

In sum, the method for correcting pump model provided by the present disclosure determines a corrected pump model based on the original pump model and the information of a zero-flow-rate power or an actual operating point at one frequency. Compared to the original pump model, the corrected pump model can be more close to the actual characteristics of the pump. Accordingly, the corrected pump model can be utilized to estimate the state of the pump to provide more accurate results.

BRIEF DESCRIPTION OF THE DRAWINGS

To make the objectives, features, advantages, and embodiments of the present disclosure, including those mentioned above and others, more comprehensible, descriptions of the accompanying drawings are provided as follows.

FIG. 1 illustrates a flowchart of a method for correcting pump model in accordance with an embodiment of the present disclosure;

FIG. 2 illustrates a schematic diagram of the Q-P curves and the corrected Q-P curves mentioned in the flowchart of FIG. 1;

FIG. 3 illustrates a schematic diagram of the Q-H curves and the corrected Q-H curves mentioned in the flowchart of FIG. 1;

FIG. 4 illustrates a schematic diagram of the Q-H-P surface mentioned in the flowchart of FIG. 1;

FIG. 5 illustrates a schematic diagram of the estimated operating points and the fitted demand curve mentioned in the flowchart of FIG. 1;

FIG. 6 illustrates a schematic diagram of the step of updating the power error of the method shown in FIG. 1;

FIG. 7 illustrates a flowchart of a method for determining head error in accordance with an embodiment of the present disclosure; and

FIG. 8 illustrates a schematic diagram of one stage of the method shown in FIG. 7.

DETAILED DESCRIPTION

For the completeness of the description of the present disclosure, reference is made to the accompanying drawings and the various embodiments described below. Various features in the drawings are not drawn to scale and are provided for illustration purposes only. To provide full understanding of the present disclosure, various practical details will be explained in the following descriptions. However, a person with an ordinary skill in relevant art should realize that the present disclosure can be implemented without one or more of the practical details. Therefore, the present disclosure is not to be limited by these details.

Reference is made to FIG. 1, which illustrates a flowchart of a method for correcting pump model 100 in accordance with an embodiment of the present disclosure. The method for correcting pump model 100 determines a corrected pump model based on the original pump model and the information of a zero-flow-rate power or an actual operating point at one frequency. Compared to the original pump model, the corrected pump model can be more close to the actual characteristics of the pump. Accordingly, the corrected pump model can be utilized to more accurately estimate the state of the pump in both sensor-equipped and sensorless systems. The method for correcting pump model 100 includes steps S102-S128, which are explained in detail below with reference to FIGS. 2-6.

As shown in FIGS. 1-3, step S102 includes: obtaining a pump model for a pump, the pump model including a Q-P curve and a Q-H curve, in which “Q” stands for (volumetric) flow rate, “P” stands for power and “H” stands for head, at each of a plurality of frequencies for the pump. In the illustrated embodiment, the pump model includes four Q-P curves QP1-QP4 and four Q-H curves QH1-QH4 at four different frequencies respectively.

In the illustrate embodiment, the pump is configured to operate at 60 Hz, 55 Hz, 50 Hz or 45 Hz. The Q-P curves QP1-QP4 are Q-P curves for the pump at 60 Hz, 55 Hz, 50 Hz or 45 Hz respectively. The Q-H curves QH1-QH4 are Q-H curves for the pump at 60 Hz, 55 Hz, 50 Hz or 45 Hz respectively. In some embodiments, the Q-P curves QP1-QP4 are substantially straight lines. In other words, each of the Q-P curves QP1-QP4 may be expressed as P=A1*Q+A0, in which the coefficient A1 is the slope of the line and the coefficient A0 is the power at zero flow rate.

In the illustrated embodiment, the unit of flow rate Q is m³/h (cubic meter per hour), the unit of power P is kW (kilowatt), and the unit of head H is m (meter). In some embodiments, the pump is a centrifugal pump driven by a frequency converter.

As shown in FIG. 1, step S104 includes: at each of the plurality of frequencies, determining a power error ΔP based on a zero-flow-rate power or an actual operating point of the pump at one of the frequencies.

For a sensorless pump system, step S104 includes: at each of the plurality of frequencies, determining the power error ΔP based on a zero-flow-rate power of the pump at one of the frequencies. In some embodiments, step S104 includes: (1) determining a first estimated flow rate of the pump based on the Q-P curve at a first frequency at which the zero-flow-rate power is obtained; (2) determining a flow rate error ratio to be a difference of the zero-flow-rate power and a P-intercept of the Q-P curve at the first frequency, divided by a slope of the Q-P curve at the first frequency, and further divided by the first estimated flow rate; (3) determining a second estimated flow rate of the pump based on the Q-P curve at a second frequency; and (4) determining the power error at the second frequency to be a product of the flow rate error ratio, the second estimated flow rate and a slope of the Q-P curve at the second frequency.

As shown in FIG. 2, considering the example where the first frequency is 60 Hz, step S104 includes: determining an estimated flow rate Q1 at the first frequency 60 Hz based on the Q-P curve QP1 and an input power P1 (e.g., input power of the frequency converter) at the first frequency 60 Hz; closing a valve of the system and obtaining a zero-flow-rate input power of the frequency converter at the first frequency 60 Hz (i.e., the zero-flow-rate power P0 of the pump); determining a flow rate error ratio QE to be a difference of the zero-flow-rate power P0 and a P-intercept A0 of the Q-P curve QP1, divided by a slope A1 of the Q-P curve QP1, and further divided by the estimated flow rate Q1 (i.e., QE=(P0−A0)/A1/Q1); determining an estimated flow rate Q4 at a second frequency 45 Hz based on the Q-P curve QP4 and an input power P4 at the second frequency 45 Hz; and determining the power error ΔP4 at the second frequency 45 Hz to be a product of the flow rate error ratio QE, the estimated flow rate Q4 and a slope A1 of the Q-P curve QP4 (i.e., ΔP4=QE*Q4*A1). The power error ΔP1 at 60 Hz, the power error ΔP2 at 55 Hz and the power error ΔP3 at 50 Hz may be determined in the same manner by choosing 60 Hz, 55 Hz or 50 Hz as the second frequency and repeating the procedure described above.

For a sensor-equipped pump system, step S104 includes: at each of the plurality of frequencies, determining the power error ΔP based on an actual operating point of the pump at one of the frequencies. In some embodiments, the actual operating point of the pump is obtained by: at a first frequency, using a pressure sensor to measure the head at the actual operating point; and determining the flow rate at the actual operating point based on the Q-H curve at the first frequency and the measured head. In some embodiments, step S104 includes: (1) determining a first estimated flow rate of the pump based on the Q-P curve at the first frequency corresponding to the actual operating point; (2) determining a flow rate error ratio to be a difference of a flow rate at the actual operating point and the first estimated flow rate, divided by the first estimated flow rate; (3) determining a second estimated flow rate of the pump based on the Q-P curve at a second frequency; and (4) determining the power error at the second frequency to be a product of the flow rate error ratio, the second estimated flow rate and a slope of the Q-P curve at the second frequency.

As shown in FIG. 2, considering the example where the first frequency is 60 Hz, step S104 includes: determining an estimated flow rate Q1 at the first frequency 60 Hz based on the Q-P curve QP1 and an input power P1 at the first frequency 60 Hz; determining a flow rate error ratio QE to be a difference of a flow rate AQ (not depicted) at the actual operating point and the estimated flow rate Q1, divided by the first estimated flow rate Q1 (i.e., QE=(AQ−Q1)/Q1); determining an estimated flow rate Q4 at a second frequency 45 Hz based on the Q-P curve QP4 and an input power P4 at the second frequency 45 Hz; and determining the power error ΔP4 at the second frequency 45 Hz to be a product of the flow rate error ratio QE, the estimated flow rate Q4 and a slope A1 of the Q-P curve QP4 (i.e., ΔP4=QE*Q4*A1). The power error ΔP1 at 60 Hz, the power error ΔP2 at 55 Hz and the power error ΔP3 at 50 Hz may be determined in the same manner by choosing 60 Hz, 55 Hz or 50 Hz as the second frequency and repeating the procedure described above.

As shown in FIG. 1, step S105 includes: determining a corrected pump model based on the pump model obtained in step S102 and the power error ΔP determined in step S104. As shown in FIGS. 2 and 3, the corrected pump model includes a corrected Q-P curve CQP1, CQP2, CQP3 or CQP4 and a corrected Q-H curve CQH1, CQH2, CQH3 or CQH4 at each of a plurality of frequencies for the pump. In some embodiments, step S105 includes steps S110-S128.

As shown in FIGS. 1 and 2, step S110 includes: at each of the plurality of frequencies, determining a corrected Q-P curve to be the Q-P curve shifted by the power error ΔP. For example, in the illustrated embodiment, the corrected Q-P curves CQP1-CQP4 are the Q-P curves QP1-QP4 shifted by the power errors ΔP1-ΔP4, respectively.

As shown in FIGS. 1 and 4, step S110 is followed by step S112, which includes: determining a Q-H-P surface QHP based on the corrected Q-P curves CQP1-CQP4 and the Q-H curves QH1-QH4 at the plurality of frequencies.

In some embodiments, step S112 includes: determining the Q-H-P surface QHP based on the corrected Q-P curves CQP1-CQP4 and the Q-H curves QH1-QH4 at the plurality of frequencies by means of surface fitting. In some embodiments, the Q-H-P surface QHP may be expressed as P=f(Q,H) (i.e., power P as a function f of flow rate Q and head H), in which f has highest degree of 3 in flow rate Q and has highest degree of 2 in head H. In some embodiments, the Q-H-P surface QHP may be expressed as P=c₀₀+c₁₀Q+c₀₁H+c₂₀Q²+c₁₁QH+c₀₂H²+c₃₀Q³+c₂₁Q²H+c₁₂QH², in which the coefficients c_(nm) may be determined based on the corrected Q-P curves CQP1-CQP4 and the Q-H curves QH1-QH4.

As shown in FIG. 1, step S112 is followed by step S114, which includes: at each of the plurality of frequencies, determining a head error h1, h2, h3 or h4 based on the Q-H-P surface QHP and the power error ΔP. In some embodiments, step S114 includes: at each of the plurality of frequencies, iteratively calculating the head error h1, h2, h3 or h4 based on the Q-H-P surface QHP and the power error ΔP (e.g., using method 200 shown in FIG. 7, which will be introduced below).

As shown in FIGS. 1 and 3, step S114 is followed by step S116, which includes: at each of the plurality of frequencies, determining a corrected Q-H curve to be the Q-H curve shifted by the head error. For example, in the illustrated embodiment, the corrected Q-H curves CQH1-CQH4 are the Q-H curves QH1-QH4 shifted by the power errors h1-h4, respectively.

After going through the steps described above to determine the corrected Q-P curves CQP1-CQP4 and the corrected Q-H curves CQH1-CQH4, in some embodiments, the method for correcting pump model 100 further includes verifying the accuracy of the corrected pump model (e.g., steps S118-S124).

As shown in FIGS. 1 and 5, in some embodiments, step S118 includes: at each of the plurality of frequencies, determining an estimated operating point F1, F2, F3 or F4 based on the corrected Q-H curve CQH1, CQH2, CQH3 or CQH4, respectively. In some embodiments, step S118 includes: determining the flow rate at the estimated operating point F1, F2, F3 or F4 based on the corrected Q-P curve CQP1, CQP2, CQP3 or CQP4 and the input power P1, P2, P3 or P4 at corresponding frequency, then determining the head at the estimated operating point F1, F2, F3 or F4 based on the determined flow rate and the corrected Q-H curve CQH1, CQH2, CQH3 or CQH4 at corresponding frequency.

As shown in FIGS. 1 and 5, step S118 is followed by step S120, which includes: determining a fitted demand curve C based on the estimated operating points F1-F4 at the plurality of frequencies. In some embodiments, step S120 includes: determining the fitted demand curve C based on the estimated operating points F1-F4 by means of regression analysis (e.g., using the method of least squares). In some embodiments, the fitted demand curve C is a quadratic curve and may be expressed as H=H_(st)+kQ², in which H_(st) is the static head, and k is the flow resistance coefficient.

As shown in FIGS. 1 and 5, step S120 is followed by step S122, which includes: determining a residual sum of squares between the estimated operating points F1-F4 at the plurality of frequencies and the fitted demand curve C. Specifically, the residual sum of squares is the sum of the squares of the head differences r (i.e, the residuals; not depicted) between the fitted demand curve C and each of the estimated operating points F1-F4. More specifically, the residual sum of squares may be expressed as Σ_(e)(r_(e))², for all estimated operating points e.

As shown in FIG. 1, step S122 is followed by step S124, which includes: determining whether the residual sum of squares exceeds an error threshold. If it is determined that the residual sum of squares does not exceed the error threshold in step S124 (i.e., the residual sum of squares is less than or equal to the error threshold), then the method for correcting pump model 100 proceeds to step S128, in which the corrected pump model is set. When the method of correcting pump model 100 reaches step S128, the process of correcting pump model is completed and the method 100 terminates.

On the other hand, if it is determined that the residual sum of squares exceeds the error threshold in step S124 (i.e., the residual sum of squares is greater than the error threshold), then the method for correcting pump model 100 further updates the corrected pump model (e.g., by proceeding to step S126).

As shown in FIG. 1, step S126 includes: updating the pump model to the corrected pump model (i.e., updating the Q-P curves of the pump model QP1-QP4 to the corrected Q-P curves CQP1-CQP4, and updating the Q-H curves QH1-QH4 of the pump model to the corrected Q-H curves CQH1-CQH4); at each of the plurality of frequencies, updating the power error ΔP based on the estimated operating point at corresponding frequency and the fitted demand curve C; after updating the power error ΔP and the pump model, performing the step of determining the corrected pump model based on the pump model and the power error ΔP (i.e., returning to step S110) to determine an updated corrected pump model. In some embodiments, the method for correcting pump model 100 repeats said procedure until the residual sum of squares converges (i.e., when the residual sum of squares is lowered to or drops below the error threshold).

As shown in FIGS. 5 and 6, in some embodiments, the step of updating the power error ΔP based on the estimated operating point and the fitted demand curve includes: (1) determining a head residual dH (not depicted) between the fitted demand curve C and the estimated operating point; determining a flow rate correction amount dQ based on the head residual dH and a differential of a function corresponding to the fitted demand curve C (i.e., differential of the function H=H_(st)+kQ²: dH=2kQdQ); and updating the power error ΔP to be a product of the flow rate correction amount dQ and a slope A1 of the corrected Q-P curve CQP.

Reference is made to FIG. 7, which illustrates a flowchart of a method 200 for determining head error in accordance with an embodiment of the present disclosure. In some embodiments, step S114 of the method for correcting pump model 100 includes the method 200 shown in FIG. 7. The method 200 uses the power error ΔP determined in step S104 or the power error ΔP updated in step S126 to iteratively correct head. The method 200 includes steps S202-S212, which are explained in detail below with reference to FIG. 8.

As shown in FIG. 7, the method 200 starts from step S202, which includes: determining a second head error ΔH based on a (first) derivative of the Q-H-P surface QHP and the power error ΔP. In some embodiments, step S202 includes: using minimum norm solution to calculate the second head error ΔH at each of the plurality of frequencies.

As shown in FIG. 7, step S202 is followed by step S204, which includes: determining a second estimated operating point based on a second Q-H curve SQH1, wherein the second Q-H curve SQH1 is the Q-H curve QH1, QH2, QH3 or QH4 shifted by the second head error ΔH at corresponding frequency. In some embodiments, the determination of the second estimated operating point includes: determining the flow rate at the second estimated operating point based on the corrected Q-P curve and the input power at corresponding frequency, then determining the head at the second estimated operating point based on the determined flow rate and the second Q-H curve SQH1. For example, FIG. 8 shows the second Q-H curve SQH1 at 60 Hz, which is the Q-H curve QH1 at 60 Hz shifted by the second head error ΔH at 60 Hz. Similarly, the second Q-H curve at 55 Hz, 50 Hz or 45 Hz can be determined by shifting the Q-H curve QH2, QH3 or QH4 by the second head error at the corresponding frequency.

Since the second head error determined ΔH in step S202 can only indicate the initial direction of correction for the Q-H curve, it is necessary to iteratively correct the second head error ΔH afterwards (steps S206-S212), such that the second Q-H curve SQH1 can be more close to the actual characteristics of the pump.

As shown in FIG. 7, step S204 is followed by step S206, which includes: determining a power difference to be a difference of an estimated power at the second estimated operating point and an input power of the pump at corresponding frequency.

In some embodiments, step S206 includes: determining a second Q-H-P surface based on the second Q-H curve SQH1; and determining the estimated power based on the second Q-H-P surface and the second estimated operating point. In some embodiments, the step of determining the second Q-H-P surface includes: determining the second Q-H-P surface based on the corrected Q-P curves CQP1-CQP4 and the second Q-H curve SQH1 by means of surface fitting. In some embodiments, the step of determining the estimated power includes: inserting the values of the flow rate Q and the head H of the second estimated operating point into the second Q-H-P surface to obtain the estimated power at corresponding frequency.

As shown in FIG. 7, step S206 is followed by step S208, which includes: determining if the power difference is within an allowable range. If it is determined that the power difference is within the allowable range in step S208, then the method 200 proceeds to step S210, in which the head error is set to the second head error. On the other hand, if it is determined that the power difference is outside the allowable range, then the method 200 proceeds to step S212, in which the second head error is updated based on the power difference, and subsequently returns to step S204. Said procedure is repeated until the power difference converges (i.e., when the power difference falls within the allowable range).

In some embodiments, the second head error may be expressed as ΔH_(n)=γ_(n)ΔH, in which ΔH is the initial second head error (i.e., the second head error determined in step S202), ΔH_(n) is the updated second head error after executing step S212 n times (i.e., the updated second head error for the n-th iteration), and γ_(n) is a value between zero and one, with the initial value γ₀ being one. Define “relative power difference” as the difference of the estimated power at the second estimated operating point and the input power of the pump at corresponding frequency, divided by the input power. The step of updating the second head error based on the power difference includes: if the absolute value of the relative power difference of the current iteration is smaller than the absolute value of the relative power difference of the previous iteration and the sign (plus or minus) of the relative power difference does not change, then decrease the value γ; if the absolute value of the relative power difference of the current iteration is larger than the absolute value of the relative power difference of the previous iteration or if the sign of the relative power difference changes, then increase the value γ.

Step S212 adjusts the amount by which the Q-H curve is to be shifted based on the power difference, such that shifted Q-H curve can be more close to the actual characteristics of the pump. As shown in FIG. 8, the absolute value of the second head error ΔH is too large such that the second Q-H curve SQH1 lies below an actual Q-H curve AQH1. Therefore, the value γ is decreased in step S212 (e.g., by choosing γ=0.5), such that the second Q-H curve SQH1 is moved towards the actual Q-H curve AQH1 (in this case, the second Q-H curve SQH1 is shifted upwards). In some embodiments, step S212 includes using bisection method to update the second head error ΔH.

It should be noted that FIG. 8 shows the actual Q-H curve AQH1 only to help explain the method 200. The method 200 itself does not involve the measurement of the actual Q-H curve AQH1.

Reference is made to the following table, which shows the procedure of iteratively calculating the second head error for the frequency of 50 Hz. Using bisection method, the relative power difference is lowered to 0.02% after five iterations. The second head error after five iterations may be used as the head error, by which the Q-H curve QH3 at 50 Hz can be shifted to obtain the corrected Q-H curve CHQ3 at 50 Hz.

n γ_(n) Relative power difference (%) 0 0 2.15 1 0.5 1.24 2 0.25 0.49 3 0.125 0.11 4 0.0625 −0.07 5 0.0938 0.02

In sum, the method for correcting pump model provided by the present disclosure determines a corrected pump model based on the original pump model and the information of a zero-flow-rate power or an actual operating point at one frequency. Compared to the original pump model, the corrected pump model can be more close to the actual characteristics of the pump. Accordingly, the corrected pump model can be utilized to estimate the state of the pump to provide more accurate results.

Although the present disclosure has been described by way of the exemplary embodiments above, the present disclosure is not to be limited to those embodiments. Any person skilled in the art can make various changes and modifications without departing from the spirit and the scope of the present disclosure. Therefore, the protective scope of the present disclosure shall be the scope of the claims as attached. 

What is claimed is:
 1. A method for correcting pump model, comprising: obtaining a pump model for a pump, the pump model comprising a Q-P curve and a Q-H curve at each of a plurality of frequencies for the pump; at each of the plurality of frequencies, determining a power error based on a zero-flow-rate power or an actual operating point of the pump at a first frequency of the plurality of frequencies; and determining a corrected pump model based on the pump model and the power error, comprising: at each of the plurality of frequencies, determining a corrected Q-P curve to be the Q-P curve shifted by the power error; determining a Q-H-P surface based on the corrected Q-P curves and the Q-H curves at the plurality of frequencies; at each of the plurality of frequencies, determining a head error based on the Q-H-P surface and the power error; and at each of the plurality of frequencies, determining a corrected Q-H curve to be the Q-H curve shifted by the head error.
 2. The method of claim 1, further comprising: at each of the plurality of frequencies, determining an estimated operating point based on the corrected Q-H curve; determining a fitted demand curve based on the estimated operating points at the plurality of frequencies; determining a residual sum of squares between the estimated operating points at the plurality of frequencies and the fitted demand curve; and if the residual sum of squares is less than or equal to an error threshold, then the corrected pump model is set.
 3. The method of claim 2, further comprising: if the residual sum of squares is greater than the error threshold, then: at each of the plurality of frequencies, updating the power error based on the estimated operating point and the fitted demand curve; at each of the plurality of frequencies, updating the Q-P curve of the pump model to the corrected Q-P curve and updating the Q-H curve of the pump model to the corrected Q-H curve; and after updating the power error and the pump model, performing the step of determining the corrected pump model based on the pump model and the power error.
 4. The method of claim 3, wherein the step of updating the power error based on the estimated operating point and the fitted demand curve comprises: determining a head residual between the fitted demand curve and the estimated operating point; determining a flow rate correction amount based on the head residual and a differential of a function corresponding to the fitted demand curve; and updating the power error to be a product of the flow rate correction amount and a slope of the corrected Q-P curve.
 5. The method of claim 1, wherein the step of determining the power error comprises: determining a first estimated flow rate of the pump based on the Q-P curve at the first frequency; determining a flow rate error ratio to be a difference of the zero-flow-rate power and a P-intercept of the Q-P curve at the first frequency, divided by a slope of the Q-P curve at the first frequency, and further divided by the first estimated flow rate; determining a second estimated flow rate of the pump based on the Q-P curve at a second frequency of the plurality of frequencies; and determining the power error at the second frequency to be a product of the flow rate error ratio, the second estimated flow rate and a slope of the Q-P curve at the second frequency.
 6. The method of claim 1, wherein the step of determining the head error comprises: at each of the plurality of frequencies, iteratively calculating the head error based on the Q-H-P surface and the power error.
 7. The method of claim 6, wherein the step of iteratively calculating the head error based on the Q-H-P surface and the power error comprises: determining a second head error based on a derivative of the Q-H-P surface and the power error; determining a second estimated operating point based on a second Q-H curve, wherein the second Q-H curve is the Q-H curve shifted by the second head error; determining a power difference to be a difference of an estimated power at the second estimated operating point and an input power of the pump at corresponding frequency; and determining the head error to be the second head error if the power difference is within an allowable range, or else updating the second head error based on the power difference and returning to the step of determining the second estimated operating point.
 8. The method of claim 7, further comprising: determining a second Q-H-P surface based on the second Q-H curve; and determining the estimated power based on the second Q-H-P surface and the second estimated operating point.
 9. The method of claim 1, wherein the step of determining the Q-H-P surface comprises: determining the Q-H-P surface based on the corrected Q-P curves and the Q-H curves at the plurality of frequencies by means of surface fitting; wherein the Q-H-P surface is expressed as P=f(Q,H), in which f has highest degree of 3 in flow rate Q and has highest degree of 2 in head H.
 10. The method of claim 1, wherein the step of determining the power error comprises: determining a first estimated flow rate of the pump based on the Q-P curve at the first frequency; determining a flow rate error ratio to be a difference of a flow rate at the actual operating point and the first estimated flow rate, divided by the first estimated flow rate; determining a second estimated flow rate of the pump based on the Q-P curve at a second frequency of the plurality of frequencies; and determining the power error at the second frequency to be a product of the flow rate error ratio, the second estimated flow rate and a slope of the Q-P curve at the second frequency. 